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Mirrors > Home > ILE Home > Th. List > hbexd | GIF version |
Description: Deduction form of bound-variable hypothesis builder hbex 1527. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
hbexd.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
hbexd.2 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
hbexd | ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbexd.1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | hbexd.2 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
3 | 1, 2 | eximdh 1502 | . 2 ⊢ (𝜑 → (∃𝑦𝜓 → ∃𝑦∀𝑥𝜓)) |
4 | 19.12 1555 | . 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓) | |
5 | 3, 4 | syl6 29 | 1 ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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